Narrow Band Omnidirectional Reflectors And Their Use As Structural Colors

ABSTRACT

Disclosed is a multilayer structure wherein a first layer of a first material having an outer surface and a refracted index between 2 and 4 extends across an outer surface of a second layer having a refractive index between 1 and 3. The multilayer stack has a reflective band of less than 200 nanometers when viewed from angles between 0° and 80° and can be used to reflect a narrow range of electromagnetic radiation in the ultraviolet, visible and infrared spectrum ranges. In some instances, the reflection band of the multilayer structure is less than 100 nanometers. In addition, the multilayer structure can have a quantity defined as a range to mid-range ratio percentage of less than 2%.

FIELD OF THE INVENTION

This invention relates generally to reflectors and methods for makingreflectors. More specifically, the invention relates to omnidirectionalreflectors and methods for making omnidirectional reflectors.

BACKGROUND OF THE INVENTION

A pigment appears as a particular color because it selectively reflectsand absorbs certain wavelengths of light. When white light, i.e. lighthaving a roughly equal mixture of the entire visible spectrum ofwavelengths, encounters a pigment, some wavelengths are absorbed by thechemical bonds and substituence of the pigment and other wavelengths arereflected. The reflected wavelengths determine the color of the pigment.This type of coloring mechanism is based on light absorption and themolecular structure generally reflects a broad range of wavelength witha moderate reflectivity (50-60%). In contrast, nature providesmagnificent colors and metal-type reflectors in insects, butterflies,birds and fish. Such colors found in nature are not based on pigments,but on the interference of light reflected from either a nanoscopicmultilayer structure of alternative high and low refractive indexmaterials or a regular array of nano-sized particles. These types ofnanostructure assemblies can reflect up to 100% of the incident light.

Such types of nanostructure assemblies, for example multilayerstructures, have not been exploited for providing narrow reflectionbands of electromagnetic radiation. Therefore, there is a need for amultilayer structure that provides a narrow reflection band, and withthe reflection band being constant when the multilayer structure isviewed from various viewing angles. As will be explained hereinbelow,the present invention provides for a multilayer structure which may beapplied to produce an omnidirectional structural color and/or anomnidirectional narrow band reflector in the visible electromagneticrange. Also explained is a method for making the multilayer structure.These and other advantages of the invention will be apparent from thedrawings and discussion presented herein.

SUMMARY OF THE INVENTION

Disclosed is a multilayer structure wherein a first layer of a firstmaterial having an outer surface and a refracted index between 2 and 4extends across an outer surface of a second layer having a refractiveindex between 1 and 3. The multilayer stack has a reflective band ofless than 200 nanometers when viewed from angles between 0° and 80° andcan be used to reflect a narrow range of electromagnetic radiation inthe ultraviolet, visible and infrared spectrum ranges. In someinstances, the reflection band of the multilayer structure is less than100 nanometers. In addition, the multilayer structure can have aquantity defined as a range to mid-range ratio of less than 2%.

In an embodiment of the present invention, the multilayer structure canbe in the form of a flake. The flake can have an average thickness rangeof between 0.5 to 5 microns and an average diameter range of between 5and 50 microns. In some instances, a plurality of flakes can be combinedwith a binder in order to form a coating material that can be used tocoat a structure. The coating material exhibits a structural color thatremains constant when viewed from various viewing angles. The pluralityof flakes of the present invention can also be applied to a structureusing other methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a multilayer structure of the presentinvention;

FIG. 2 is a graphical representation of bandedge as a function ofincident angle;

FIG. 3 is a graphical representation comparing an exact solution and anapproximate solution for bandedge as a function of incident angle;

FIG. 4A is a graphical representation of range to mid-range ratios forthe transverse magnetic mode of electromagnetic radiation;

FIG. 4B is a graphical representation of range to mid-range ratios forthe transverse electric mode of electromagnetic radiation;

FIG. 5A is a graphical representation of range to mid-range ratios equalto 30% and 0.2%;

FIG. 5B is a graphical representation of corresponding reflectancespectra for the range to mid-range ratios of 30% and 0.2% shown in FIG.5A;

FIG. 6 is a graphical representation showing a comparison of the rangeto mid-range ratio of 0.2% for the transverse magnetic mode andtransverse electric mode of electromagnetic radiation;

FIG. 7A is a graphical representation of the reflectance as a functionof wavelength for Case I shown in FIG. 6;

FIG. 7B is a graphical representation of the reflectance as a functionof wavelength for Case II shown in FIG. 6;

FIG. 7C is a graphical representation of reflectance as a function ofwavelength for Case III shown in FIG. 6;

FIG. 7D is a graphical representation of the dispersion of the centerwavelength in Case I, II and III;

FIG. 8 is a graphical representation of the comparison of approximateand exact solutions for the bandedges of a multilayer structure designedaccording to the quarter wave technique;

FIG. 9A is a graphical representation of a center wavelength dispersionfactor as a function of high refractive indices and low refractiveindices;

FIG. 9B is a graphical representation of the range to mid-range ratiosfor transverse electric mode and traverse magnetic mode wherein adesired region of high reflective indices and low reflective indices ishighlighted;

FIG. 9C is a graphical representation of narrow band omnidirectionalreflectivity of a case with low refractive index contrast between highand low refractive index material;

FIG. 9D is a graphical representation of band structure of a narrow bandomnidirectional reflective design with low refractive index contrastbetween high and low refractive index material.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention includes a multilayer omnidirectional reflectorthat retains a specific reflection band of ultraviolet, visible orinfrared magnetic radiation from arbitrary angles of incidence. As such,the present invention has utility as an omnidirectional reflector for anarrow wavelength range of electromagnetic radiation. In addition, thepresent invention includes a method for the making of theomnidirectional reflector.

The omnidirectional reflector of the present invention is a multilayeredbody having a first layer with a first refractive index and a secondlayer with a second refractive index. In some instances the differencebetween the refractive indices of the two layers can range between 0.2to 1.0, and the multilayer structure has a reflection band of less than200 nanometers when viewed from angles between 0° and 80°. In otherinstances the difference between the refractive indices of the twolayers can range between 0.2 and 0.6, and the multilayer structure has areflection band of less than 100 nm when viewed from angles between 0°and 65°.

Referring now to FIG. 1, there is shown a multilayer structure 10 havingalternating layers of a first material 100 with a high refractive index(n_(H)) and a thickness (d_(H)), and a second material 200 with a lowrefractive index (n_(L)) and a thickness (d_(L)). The first material 100includes and outer surface 110 that can extend across an outer surface210 of the second material 200. In some instances, the multilayerstructure 10 has a total number of layers greater than three. In otherinstances, the multilayer structure 10 has a total number of layersgreater than seven.

An electromagnetic wave consisting of perpendicular electric (E) andmagnetic (M) vector components is shown incident to the multilayerstructure at an incident angle θ₀. The electromagnetic wave can bedistinguished into two independent electromagnetic modes: a transverseelectric (TE) mode and a transverse magnetic (TM) mode. The refractiveindex of the medium beyond the multilayer structure 10 at a first end 12is n₀. For example, when the medium is air, n₀=1. The refractive of anoptional substrate at a second end 14 is n_(Substrate). The optionalsubstrate can be any material compatible with the multilayer structure10 and call assist in the manufacture, storage, shipping and/or handlingof the structure. If an optional substrate is present, it may or may notbe removed after the manufacture of the multilayer structure 10.

When electromagnetic radiation impacts a material surface, waves of theradiation can be reflected from or transmitted through the material.Furthermore, when electromagnetic radiation impacts the first end 12 ofthe multilayer structure 10 at the angle θ₀, the reflected angles theelectromagnetic waves make with the surface of the high and lowrefractive index layers are θ_(H) and θ_(L), respectively. Using Snell'slaw:

n₀ Sin θ₀=n_(L) Sin θ_(L)=n_(H) Sin θ_(H)   (1)

the angles θ_(H) and θ_(L) can be determined if the refractive indicesn_(H) and n_(L) are known.

Regarding omnidirectional reflectivity, a necessary but not sufficientcondition for the TE mode and the TM mode of electromagnetic radiationrequires the maximum angle of refraction (θ_(H,MAX)) inside the firstlayer to be less than the Brewster angle (θ_(B)) of tie interfacebetween the first layer and the second layer. If this condition is notsatisfied, the TM mode of the electromagnetic waves will not bereflected at the second and all subsequent interfaces and thus willtransmit through the structure. Using this consideration:

$\begin{matrix}{{{Sin}\; \theta_{H,{Max}}} = \frac{n_{0}}{n_{H}}} & (2) \\{and} & \; \\{{{Tan}\; \theta_{B}} = \frac{n_{L}}{n_{B}}} & (3)\end{matrix}$

Thereby requiring:

$\begin{matrix}{n_{0} < \frac{n_{H}n_{L}}{\sqrt{n_{H}^{2} + n_{L}^{2}}}} & (4)\end{matrix}$

In addition to the necessary condition represented by Equation 4, ifelectromagnetic radiation of wavelength λ falls on a multilayerstructure with an angle θ₀, and the individual bi-layers of themultilayer structure have thicknesses d_(H) and d_(L) with respectiverefractive indices n_(H) and n_(L), the characteristic translationmatrix (F_(T)) call be expressed as:

$\begin{matrix}{F_{T} = {\frac{1}{1 + \rho_{T}}{\begin{matrix}^{{\delta}_{L}} & {\rho_{T}^{{\delta}_{L}}} \\{\rho_{T}^{{\delta}_{L}}} & ^{- {\delta}_{L}}\end{matrix}} \times \frac{1}{1 - \rho_{T}}{\begin{matrix}^{{\delta}_{H}} & {\rho_{T}^{- {\delta}_{H}}} \\{\rho_{T}^{{\delta}_{H}}} & ^{- {\delta}_{H}}\end{matrix}}}} & (5)\end{matrix}$

which can also be expressed as:

$\begin{matrix}{F_{T} = {\frac{1}{1 - \rho_{T}^{2}}{\begin{matrix}{^{{({\delta_{L} + \delta_{n}})}} - {\rho_{T}^{2}^{- {{({\delta_{u} - \delta_{L}})}}}}} & {{- 2}\; \rho_{T}^{- _{u}}{Sin}\; \delta_{L}} \\{2{\rho}_{T}^{{\delta}_{u}}{Sin}\; \delta_{L}} & {^{{({\delta_{L} + \delta_{u}})}} - {\rho_{T}^{2}^{- {{({\delta_{H} - \delta_{L}})}}}}}\end{matrix}}}} & (6)\end{matrix}$

and where:

$\begin{matrix}{\delta_{H} = {\frac{2\pi}{\lambda}n_{H}d_{H}{Cos}\; \theta_{H}}} & (7) \\{\delta_{L} = {\frac{2\pi}{\lambda}n_{L}d_{L}{Cos}\; \theta_{L}}} & (8) \\{{{Cos}\; \theta_{H}} = \sqrt{1 - \frac{n_{o}^{2}{Sin}^{2}\theta_{0}}{n_{H}^{2}}}} & (9) \\{and} & \; \\{{{Cos}\; \theta_{L}} = \sqrt{1 - \frac{n_{o}^{2}{Sin}^{2}\theta_{0}}{n_{L}^{2}}}} & (10)\end{matrix}$

In addition,

$\begin{matrix}{\rho_{T} = \frac{n_{HT} - n_{LT}}{n_{HT} + n_{LT}}} & (11) \\{where} & \; \\{n_{HT} = \{ \begin{matrix}\frac{n_{L}}{{Cos}\; \theta_{H}} \\{n_{H}{Cos}\; \theta_{d}}\end{matrix} } & (12) \\( {{for}\mspace{14mu} {TM}\mspace{14mu} {and}\mspace{14mu} {TE}\mspace{14mu} {polarization}\mspace{14mu} {respectively}} ) & \; \\{n_{LT} = \{ \begin{matrix}\frac{n_{L}}{{Cos}\; \theta_{L}} \\{n_{L}{Cos}\; \theta_{L}}\end{matrix} } & (13) \\( {{for}\mspace{14mu} {TM}\mspace{14mu} {and}\mspace{14mu} {TE}\mspace{14mu} {polarization}\mspace{14mu} {respectively}} ) & \;\end{matrix}$

Solving ρ_(T) explicitly for TE and TM:

$\begin{matrix}{\rho_{TM} = \frac{{n_{H}{Cos}\; \theta_{L}} - {n_{L}{Cos}\; \theta_{H}}}{{n_{H}{Cos}\; \theta_{L}} + {n_{L}{Cos}\; \theta_{H}}}} & (14) \\{and} & \; \\{\rho_{TE} = \frac{{n_{H}{Cos}\; \theta_{H}} - {n_{L}{Cos}\; \theta_{L}}}{{n_{H}{Cos}\; \theta_{H}} + {n_{L}{Cos}\; \theta_{L}}}} & (15)\end{matrix}$

A viewing angle dependant band structure can be obtained from a boundarycondition for the edge, also known as the bandedge, of the totalreflection zone. For the purposes of the present invention, bandedge isdefined as the equation for the line that separates the total reflectionzone from the transmission zone for the given band structure.

A boundary condition that determines the bandedge frequencies of thehigh reflectance band can be given by:

$\begin{matrix}{{{Trace}\; {F_{T}}} = {- 1}} & (16)\end{matrix}$

Thus, from equation 3:

$\begin{matrix}{\frac{{{Cos}( {\delta_{H} + \delta_{H}} )} - {\rho_{T}^{2}{{Cos}( {\delta_{H} - \delta_{L}} )}}}{1 - \rho_{T}^{2}} = {- 1}} & (17)\end{matrix}$

or expressed differently:

$\begin{matrix}{{{Cos}^{2}( \frac{\delta_{H} + \delta_{L}}{2} )} = {\rho_{T}^{2}{{Cos}^{2}( \frac{\delta_{H} - \delta_{L}}{2} )}}} & (18)\end{matrix}$

Combining equations 15 and 7, the following bandedge equation isobtained:

$\begin{matrix}{{{Cos}( \frac{\pi \; L_{+}}{\lambda} )} = {{\pm {\rho_{T}}}{{Cos}( \frac{\pi \; L_{-}}{\lambda} )}}} & (19)\end{matrix}$

Where:

L ₊ =n _(H) d _(H) Cos θ_(H) +n _(L) d _(L) Cos θ_(L)   (20)

and:

L ⁻ =n _(H) d _(H) Cos θ_(H) −n _(L) d _(L) Cos θ_(L)   (21)

The + sign in the bandedge equation shown above represents the bandedgefor the long wavelength (λ_(long)) and the − sign represents thebandedge for the short wavelength (λ_(short)). Recompiling equations 20and 21:

$\begin{matrix}{{{{Cos}( \frac{\pi \; L_{+}}{\lambda_{long}} )} = {{+ {\rho_{TE}}}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{long}} )}}}{and}{{{Cos}( \frac{\pi \; L_{+}}{\lambda_{Short}} )} = {{- {\rho_{TE}}}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Short}} )}}}} & (22)\end{matrix}$

for the TE mode, and:

$\begin{matrix}{{{{Cos}( \frac{\pi \; L_{+}}{\lambda_{long}} )} = {{+ {\rho_{TM}}}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{long}} )}}}{and}{{{Cos}( \frac{\pi \; L_{+}}{\lambda_{Short}} )} = {{- {\rho_{TM}}}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Short}} )}}}} & (23)\end{matrix}$

for the TM mode.

An approximate solution of the bandedge can be determined by thefollowing expression:

L ⁻ =n _(H) d _(H) Cos θ_(H) −n _(L) d _(L) Cos θ_(L)˜0   (24)

This approximate solution is reasonable when considering a quarter wavedesign (described in greater detail below) and optical thicknesses ofthe alternating layers chosen to be equal to each other. In addition,relatively small differences in optical thicknesses of the alternatinglayers provide a cosine close to unity. Thus, equations 23 and 24 yieldapproximate bandedge equations:

$\begin{matrix}{{{\lambda_{long}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{\rho_{TE}( \theta_{0} )}}}}{and}{{\lambda_{Short}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}( {- {{\rho_{TE}( \theta_{0} )}}} )}}} & (25)\end{matrix}$

for the TE mode and:

$\begin{matrix}{{{\lambda_{long}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{\rho_{TM}( \theta_{0} )}}}}{and}{{\lambda_{Short}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}( {- {{\rho_{TM}( \theta_{0} )}}} )}}} & (26)\end{matrix}$

for the TM mode.

Values for L₊ and ρ_(TM) as a function of incident angle can be obtainedfrom equations 7, 8, 14, 15 90 and 21, thereby allowing calculations forλ_(long) and λ_(short) in the TE and TM modes as a function of incidentangle.

Turning to FIG. 2, the TE and TM bandedges as a function of incidentangle on a multilayer system with a first material having a highrefractive index equal to 4.6 and a thickness of 800 nanometers and asecond layer material with a refractive index of 1.6 and a thickness of1600 nanometers are shown. The omnidirectional band is defined by thewavelength range where electromagnetic radiation coming from any anglewill be completely reflected as shown by the highlighted box. For theexample shown in FIG. 2, the omnidirectional band is in the infraredregion and is approximately between the wavelengths of 9.34 microns and15 microns. Mathematically, the omnidirectional band shown in FIG. 2 canbe written as:

Δλ_(omni)=λ_(long) ^(TM)(90°)−λ_(short) ^(TE)(0°)   (27)

An exact solution to the bandedge equations of equation 23 and equation24 can be represented as:

$\begin{matrix}{{{\lambda_{long}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{{\rho_{TE}( \theta_{0} )}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Long}} )}}}}}{and}{{\lambda_{Short}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{{\rho_{TE}( \theta_{0} )}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Short}} )}}}}}} & (28)\end{matrix}$

for the TE mode, and:

$\begin{matrix}{{{\lambda_{long}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{{\rho_{TM}( \theta_{0} )}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Long}} )}}}}}{and}{{\lambda_{Short}( \theta_{0} )} = \frac{\pi \; {L_{+}( \theta_{0} )}}{{Cos}^{- 1}{{{\rho_{TM}( \theta_{0} )}{{Cos}( \frac{\pi \; L_{-}}{\lambda_{Short}} )}}}}}} & (29)\end{matrix}$

for the TM mode. Using numerical evaluation, a comparison between theexact and approximate solutions for the multilayer system describedabove is shown in FIG. 3. FIG. 3 thus demonstrates that an approximationmethod for the determination of the bandedge equations is reasonable andadequate.

The center wavelength of an omnidirectional reflector (λ_(c)), can bedetermined from the relation:

λ_(c=)2(n _(H) d _(H) Cos θ_(H) +n _(L) d _(L) Cos θ_(L))   (30)

The center wavelength can be an important parameter since its valueindicates the approximate range of electromagnetic wavelength and/orcolor spectrum to be reflected. For example, the multilayer systemdescribed above for normal incidence provides a center wavelength of12.5 microns, which is consistent with the plots shown in FIGS. 2 and 3.

Another important parameter that can provide an indication as to thewidth of a reflection band is defined as the ratio of range ofwavelengths within the omnidirectional reflection band to the mid-rangeof wavelengths within the omnidirectional reflection band. This “rangeto mid-range ratio” (η) is mathematically expressed as:

$\begin{matrix}{\eta_{TE} = {2\frac{{\lambda_{avg}^{TE}( {\theta_{0} = 90^{{^\circ}}} )} - {\lambda_{Short}^{TE}( {\theta_{0} = 0^{{^\circ}}} )}}{{\lambda_{avg}^{TE}( {\theta_{0} = 90^{{^\circ}}} )} + {\lambda_{Short}^{TE}( {\theta_{0} = 0^{{^\circ}}} )}}}} & (31)\end{matrix}$

for the TE mode, and:

$\begin{matrix}{\eta_{TM} = {2\frac{{\lambda_{avg}^{TM}( {\theta_{0} = 90^{{^\circ}}} )} - {\lambda_{Short}^{TM}( {\theta_{0} = 0^{{^\circ}}} )}}{{\lambda_{avg}^{TM}( {\theta_{0} = 90^{{^\circ}}} )} + {\lambda_{Short}^{TM}( {\theta_{0} = 0^{{^\circ}}} )}}}} & (32)\end{matrix}$

for the TM mode. It is appreciated that the range to mid-range ratio canbe expressed as a percentage and for the purposes of the presentinvention, the term range to mid-range ratio and range to mid-rangeratio percentage are used interchangeably. It is further appreciatedthat a ‘range to mid-range ratio’ value provided herein having a ‘%’sign following is a percentage value of the range to mid-range ratio.The range to mid-range ratios for the TM mode and TE mode can benumerically calculated from equations 31 and 32 and plotted as afunction of high refractive index and low refractive index, asillustrated in FIGS. 4A and 4B. Furthermore, once the range to mid-rangeratio has been determined, corresponding reflectance as a function ofwavelength can be plotted.

An example of the reflectance as a function of the range to mid-rangeratio is demonstrated in FIGS. 5A and 5B. FIG. 5A shows two curves for aTM mode range to mid-range ratio—one for η_(TM) equal to 0.2% and onefor η_(TM) equal to 30%. FIG. 5B shows the corresponding reflectance forrange to mid-range ratios labeled ‘A’ and ‘B’ in FIG. 5A with angles ofincidence ranging from 0° to 45°. With a range to mid-range ratio of 30%and the angles of incidence ranging from 0° to 45°, the reflection bandillustrated in FIG. 5B is approximately 300 nanometers. In contrast, fora range to mid-range ratio of 0.2% and the same angles of incidence, thereflection band is approximately 100 nanometers.

Regarding the center wavelength of the omnidirectional reflector,equation 30 demonstrates that the center wavelength, and therefore thedispersion of the center wavelength, is a function of the incidenceangle. In some instances, the omnidirectional reflectors of the presentinvention have a small dispersion of the center wavelength as a functionof the incidence angle.

The narrower the range of the dispersion of the center wavelength, thepurer the observed color since a more narrow band of wavelengths arereflected from the reflector to, for example, a human eye.

A method to control the dispersion of the center wavelength can includecomparison of the range to mid-range ratios for the TM mode and the TEmode as a function of high reflection indices and low reflectionindices. FIG. 6 illustrates a range to mid-range ratio of 0.2% for theTM mode and the TE mode as a function of high refractive index and lowrefractive index. As illustrated in FIG. 6, a relatively largedifference between the high refractive indices for the TM mode and TEmode is shown by Case I, an intermediate difference by Case II, and arelatively snip difference by Case III. Thus for a given range tomid-range ratio, different values for the high refractive index and thelow refractive index can be chosen.

Turning to FIG. 7A, the reflectance as a function of wavelength for CaseI is illustrated wherein the high refractive index equals 2.61, the lowrefractive index equals 1.2 and the angle of incidence ranges from 0° to45°. As illustrated by this figure, the center wavelength shiftssignificantly when electromagnetic radiation incident normal to themultilayer structure is compared to electromagnetic radiation incident45° to the structure. In contrast a relatively small difference betweenthe high refractive index and the low refractive index, and equivalentangles of incidence, results in a small dispersion of the centerwavelength as shown in FIG. 7C. Thus, for a narrow range of wavelengthsto be reflected by a multilayer structure, a relatively small differencebetween the refractive indices between the first material 100 and thesecond material 200 is desired. FIG. 7D quantifies the dispersion incenter wavelength with varying incident angle for Case I, II and III,and illustrates the reduction in dispersion from approximately 140 nmfor Case I to approximately 40 nm for Case III.

In another embodiment of the present invention, a quarter wave techniquecan be used to determine the refractive indices and thicknesses ofalternating layers of material for an omnidirectional reflector. Usingthis method, the optical thicknesses of the high refractive indexmaterial and low refractive index material are set to be equal to eachother, and equal to one-fourth of a desired reflective wavelength. Thus,once the refractive indices of the multilayer structure have beenselected, the thicknesses of the individual layers are set based on thefollowing equation:

$\begin{matrix}{{\eta_{H}d_{H}} = {{\eta_{L}d_{L}} = \frac{\lambda_{o}}{4}}} & (33)\end{matrix}$

where λ₀=λ_(c) at θ₀=0.

Turning to FIG. 8, a graphical representation of an approximate solutionto the bandedges of a quarter wave omnidirectional reflector is shownaccording to the parameters of Case II mentioned above. This figure alsoshows the exact solutions whereby similar results are obtained. Asillustrated in this figure, a narrow omnidirectional band at 490nanometers is consistent with the reflectance band shown in FIG. 7B. Itis appreciated that to obtain the narrow omnidirectional band that thedispersion of the center wavelength must be minimized. Thus, fromequation 30, the dispersion of the center wavelength can be expressedas:

$\begin{matrix}\begin{matrix}{{\Delta \; \lambda_{c}} = {\lambda_{c}_{\theta_{0} = 0^{{^\circ}}}{{- \lambda_{c}}_{\theta_{0} = 90^{{^\circ}}}}}} \\{= {2( {\frac{n_{H}d_{H}}{1} + \frac{n_{L}d_{L}}{1} - \frac{n_{H}d_{H}}{\sqrt{1 - \frac{n_{0}^{2}}{n_{H}^{2}}}} - \frac{n_{L}d_{L}}{\sqrt{1 - \frac{n_{0}^{2}}{n_{L}^{2}}}}} )}}\end{matrix} & (34)\end{matrix}$

where:

$\begin{matrix}{{\Delta \; \lambda_{c}} = {\frac{\lambda_{0}}{4}F_{c}}} & (35)\end{matrix}$

and F_(c), the center wavelength dispersion factor can be expressed as:

$\begin{matrix}{F_{c} = ( {2 - \frac{1}{\sqrt{1 - \frac{n_{0}^{2}}{n_{n}^{2}}}} - \frac{1}{\sqrt{1 - \frac{n_{0}^{2}}{n_{L}^{2}}}}} )} & (36)\end{matrix}$

The center wavelength dispersion factor is shown in FIG. 9A as afunction of the high refractive index and the low refractive index.Thus, from equation 36 and FIG. 9A, the dispersion of the centerwavelength can be reduced with the proper selection of high and lowrefractive index material. Also illustrated in FIG. 9A with the “WideBands” arrow is the fact that a multilayer structure exhibiting a largedifference between the high refractive index and the low refractiveindex will possess a wide reflection band even though the centerwavelength dispersion factor is relatively low. Likewise, when thealternating layers possess a first material with a high refractive indexmaterial that approaches the low refractive index of the secondmaterial, higher side bands of the reflected electromagnetic radiationoccur as illustrated by the “High side bands” arrow. The higher sidebands can be reduced using a variety of methods, illustrativelyincluding the use of Rugate filters.

FIG. 9B provides a targeted region for range to mid-range ratios, highrefractive indices and low refractive indices. When the differencebetween range to mid-range ratio of the TE and TM modes is relativelylarge, a wide or large reflection band of the multilayer structureoccurs. In contrast, for relatively small range to mid-range ratios, arelatively narrow reflection band is exhibited with a targeted regime ofsuch values shown in the diagram. In addition, FIGS. 9C and 9Dillustrate that when a small refractive index contrast (0.39) is chosenbetween the high and low refractive index materials, a narrow bandwidthomnidirectional reflector is obtained in the visible region.

Thus, in some instances, omnidirectional reflectors have alternatinglayers of materials wherein one material has a low refractive indexrange between 1 and 3 and another material has a high refractive indexrange between 2 and 4. In addition, the difference between the lowrefractive index material and the high refractive index material, hereindefined as the refractive index contrast, ranges between 0.2 and 1.0,and the range to mid-range ratio percentage varies from a value greaterthan zero to 10%. In other instances, the materials used for thealternating layers of an omnidirectional reflector include a firstmaterial with a low refractive index ranging between 2 and 3, a secondmaterial with a high refractive index ranging from 2.47 to 3.38. In yetother instances, the difference between the high refractive indexmaterial and the low refractive index material can be between 0.35 and0.5, and/or the range to mid-range ratio percentage can be a valuegreater than zero and 5%. In some instances the mid-range ratiopercentage can also range between a value greater than zero and 2%.Suitable materials for the production of an omnidirectional reflector ofthe present invention are chosen such that the above-stated criteria aremet.

Table 1 shows possible, but not limited to, high and low refractiveindex materials, respectively, for an omnidirectional reflectorexhibiting a narrow reflective band. Thus, by choosing appropriatematerials such that the difference between the refractive indices isbetween 0.2 to 1.0, and the range to mid-range ratio percentage isbetween a positive value greater than zero and 10%, an omnidirectionalreflector which affords for a structural color that remains constantwhen viewed from various angles is provided. In some instances thestructural color provided by the omnidirectional reflector of thepresent invention remains constant when view from angles between 0° to80°. In other instances, the structural color remains constant whenviewed from angles between 0° to 65°. In yet other instances, thestructural color remains constant when viewed from angles between 0° to45°.

Again, Table 1 is for illustrative purposes only and in no way limitsthe scope of the present invention. Any two layers having a differencebetween the refractive indices between 0.2 to 1.0, and a range tomid-range ratio percentage between a positive value greater than zeroand 10% is within the scope of the present invention. In addition, it iswithin the scope of the present invention that more than two differentmaterials can be used in the multilayer stack and/or that one of thealternating layers can be a defect layer, i.e. a layer of material withdefects purposefully therein in order to obtain a desired refractiveindex.

It is appreciated that the omnidirectional reflector of the presentinvention can be in the form of particles, discs, flakes, and the like.Furthermore, it is also appreciated that the particles, discs, and/orflakes can be mixed with suitable organic and/or inorganic binders inorder to form a coating. The binder and the omnidirectional reflector ofthe present invention can thus be used to provide a paint and/or coatinghaving a color that does not change when viewed from various viewingangles. In addition, the particles, discs, and/or flakes of the presentinvention can be applied to a surface using other methods, such aselectrostatic charging, e-coating, powder coating, spray deposition andthe like, such that a surface provides a color that does not change inappearance when viewed from a variety of viewing angles.

TABLE 1 Refractive Index Materials Refractive Index Materials (visibleregion) (visible region) Refrac- Refrac- tive tive Material IndexMaterial Index Germanium (Ge) 4.0-5.0 Chromium (Cr) 3.0 Tellurium (Te)4.6 Tin Sulfide (SnS) 2.6 Gallium Antimonite 4.5-5.0 Low Porous Si 2.56(GaSb) Indium Arsenide (InAs) 4.0 Chalcogenide glass 2.6 Silicon (Si)3.7 Cerium Oxide (CeO₂) 2.53 Indium Phosphate (InP) 3.5 Tungsten (W) 2.5Gallium Arsenate (GaAs) 3.53 Gallium Nitride (GaN) 2.5 Gallium Phosphate(GaP) 3.31 Manganese (Mn) 2.5 Vanadium (V) 3 Niobium Oxie (Nb₂O₃) 2.4Arsenic Selenide 2.8 Zinc Telluride (ZnTe) 3.0 (As₂Se₃) CuAlSe₂ 2.75Chalcogenide glass + Ag 3.0 Zinc Selenide (ZnSe) 2.5-2.6 Zinc Sulfate(ZnSe) 2.5-3.0 Titanium Dioxide 2.36 Titanium Dioxide (TiO₂) - 2.43(TiO₂) - vacuum deposited solgel Alumina Oxide (Al2O3) 1.75 SodiumAluminum Fluoride 1.6 (Na3AlF6) Yttrium Oxide (Y2O3) 1.75 PolyetherSulfone (PES) 1.55 Polystyrene 1.6 High Porous Si 1.5 Magnesium Fluoride1.37 Indium Tin Oxide nanorods 1.46 (MgF2) (ITO) Lead Fluoride (PbF2)1.6 Lithium Fluoride (LiF4) 1.45 Potassium Fluoride (KF) 1.5 CalciumFluoride 1.43 Polyethylene (PE) 1.5 Strontium Fluoride (SrF2) 1.43Barium Fluoride (BaF2) 1.5 Lithium Fluoride (LiF) 1.39 Silica (SiO2) 1.5PKFE 1.6 PMMA 1.5 Sodium Fluoride (NaF) 1.3 Aluminum Arsenate 1.56Nano-porous Silica (SiO2) 1.23 (AlAs) Solgel Silica (SiO2) 1.47Sputtered Silica (SiO2) 1.47 N,N′ bis(l naphthyl)- 1.7 Vacuum DepositedSilica 1.46 4,4′Diamine (NPB) (SiO2) Polyamide-imide (PEI) 1.6

A flake of the present invention can have an average thickness between0.5 and 5 microns and an average diameter between 5 and 50 microns. Forthe purpose of the present invention, the term average thickness isdefined as the average value taken from at least three differentthickness measurements and the term average diameter is defined as theaverage value taken from at least three different diameter measurements.It is appreciated that the flake can have an optional substrate attachedthereto or be a freestanding flake. The substrate can be made from anymaterial known to those skilled in the art, illustratively includingmetals, alloys, plastics, ceramics, glasses and combinations thereof,and may or may not be removable after the flake is produced.

It is appreciated that narrow band omnidirectional reflectors of thepresent invention can also be designed, manufactured and used to reflectultra violet (UV) light. Thus, narrow band omnidirectional reflectors ofthe present invention can be used to produce UV-reflecting coatingswherein UV-reflecting narrow band omnidirectional reflectors made areadded to: (1) currently available paints, stains and the like: (2)coatings of the present invention containing narrow band omnidirectionalreflectors which provide visible color; and/or (3) suitable clearbinder(s) to produce a clear coating having UV protection capabilities.It is also appreciated that narrow band omnidirectional reflectors ofthe present invention can be used in telecommunication andoptoelectronic devices.

Methods for producing the omnidirectional reflector of the presentinvention include the sol gel process, electron gun evaporation ofalternating layers, vacuum evaporation of alternating layers, thermalevaporation, CVD processes, electrochemical deposition and etchingprocesses, high-vacuum vapor deposition and oxidation processes,sputtering of alternating layers, molecular-beam-epitaxy processes,thermal mechanical processing, chemical processing, poly-electrolytemultilayer deposition by ‘layer by layer’ processing and/or combinationsthereof.

In this manner, narrow bandwidth omnidirectional reflectors and methodsfor their production are provided. The foregoing drawings, discussionand description are illustrative of specific embodiments of the presentinvention, but they are not meant to be limitations upon the practicethereof. Numerous modifications and variations of the invention will bereadily apparent to those of skill in the art in view of the teachingspresented herein. It is the following claims, including all equivalentswhich define the scope of the invention.

1. A multilayer stack, said stack comprising: a first layer of a firstmaterial having an outer surface and a refractive index between 2 and 4;and a second layer of a second material having an outer surface and arefractive index between 1 and 3, said outer surface of said secondlayer extending across said outer surface of said first layer; saidmultilayer stack having a reflection band of less than 200 nanometerswhen viewed from angles between 0 degrees and 80 degrees.
 2. Themultilayer stack of claim 1, wherein said reflection band is less than200 nanometers when viewed from angles between 0 degrees and 65 degrees.3. The multilayer stack of claim 1, wherein said reflection band is lessthan 100 nanometers when viewed from angles between 0 degrees and 65degrees.
 4. The multilayer stack of claim 1, wherein said first layerhas a refractive index between 2.4 and 3.4 and said second layer has arefractive index between 2 and
 3. 5. The multilayer stack of claim 1,wherein the difference between said refractive index of said first layerand said refractive index of said second layer is between 0.2 and
 1. 6.The multilayer stack of claim 1, wherein the difference between saidrefractive index of said first layer and said refractive index of saidsecond layer is between 0.35 and 0.5.
 7. The multilayer stack of claim1, wherein said stack has a range to mid-range ratio percentage betweena value greater than zero and 10%.
 8. The multilayer stack of claim 1,wherein said stack has a range to mid-range ratio percentage between avalue greater than zero and 5%.
 9. The multilayer stack of claim 1,wherein said stack has a range to mid-range ratio percentage between avalue greater than zero and 2%.
 10. The multilayer stack of claim 1,wherein said stack has more than three total layers.
 11. The multilayerstack of claim 1, wherein said stack has more than seven total layers.12. The multilayer stack of claim 1, wherein said stack is in the formof a flake.
 13. The multilayer stack of claim 12 wherein said flake hasan average thickness range of between 0.5 and 5 microns.
 14. Themultilayer stack of claim 12, wherein said flake has an average diameterrange of between 5 and 50 microns.
 15. The multilayer stack of claim 12,wherein said flake is mixed with a binder to make a paint.
 16. Themultilayer stack of claim 12, wherein said flake is mixed with a binderto make a UV-protective coating.
 17. An omnidirectional reflector, saidreflector comprising: a first layer of a first material having an outersurface and a refractive index between 2 and 4; and a second layer of asecond material having an outer surface and a refractive index between 1and 3, said outer surface of said second layer extending across saidouter surface of said first layer and forming a multilayer stack; saidrefractive index of said first layer and said refractive index of saidsecond layer having a difference between 0.2 and 1; said multilayerstack having a range to mid-range ratio percentage between a positivepercentage greater than zero and 100%; said multilayer stack also havinga reflection band of less than 200 nanometers when viewed from anglesbetween 0 degrees and 80 degrees.
 18. The multilayer stack of claim 17,wherein said reflection band is less than 100 nanometers when viewedfrom angles between 0 degrees and 65 degrees.
 19. The multilayer stackof claim 17, wherein said first layer has a refractive index between 2.4and 3.4 and said second layer has a refractive index between 2 and 3.20. A narrow bandwidth omnidirectional reflector comprising: a firstlayer of a first material having an outer surface and a refractive indexbetween 2.4 and 3.4; a second layer of a second material having arefractive index between 2 and 3, said outer surface of said first layerextending across said second layer and forming a multilayer stack; saidmultilayer stack having a range to mid-range ratio percentage between apositive percentage greater than zero and 5%; said multilayer stack alsohaving a reflection band of less than 100 nanometers when viewed fromangles between 0 degrees and 65 degrees.